An analytic solution of high-beta equilibrium in a large aspect ratio tokamak

Abstract

An analytic solution of the high-beta (ε β_p~β_q²/ε»1) equilibrium of a large aspect ratio tokamak is presented. Two arbitrary flux functions, the pressure profile p(φ) and the safety factor profile q(φ), specify the equilibrium. The solution splits into two asymptotic regions: the core region where β is a function of the major radius alone and a narrow boundary layer region adjoining the conducting wall. The solutions in the two regions are asymptotically matched to each other. For monotonic pressure profiles, the Shafranov shift is equal to the minor radius. For beta much bigger than 1, the solution contains a region (in place of the magnetic axis) of zero magnetic field and constant pressure. At high beta the quantity β_I, which is essentially proportional to the pressure over the total current squared, is largely independent of pressure. The important ramifications of limited β_I for high-beta reactors are discussed. Generalizations to shaped cross sections and hollow pressure profiles are outlined. The problem of equilibrium reconstruction in the high-beta regime is also considered.